CN112068424A - Discrete repetitive control method of ellipse approximation law adopting disturbance compensation - Google Patents

Discrete repetitive control method of ellipse approximation law adopting disturbance compensation Download PDF

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CN112068424A
CN112068424A CN202010787437.XA CN202010787437A CN112068424A CN 112068424 A CN112068424 A CN 112068424A CN 202010787437 A CN202010787437 A CN 202010787437A CN 112068424 A CN112068424 A CN 112068424A
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servo system
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胡轶
朱鹏升
薛良曾
孙明轩
陈强
吴春
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Zhejiang University of Technology ZJUT
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Abstract

A discrete repetitive control method for a motor servo system by using an ellipse approximation law of disturbance compensation, wherein a given module generates periodic reference signals, a periodic feedback link is constructed, the ellipse approximation law is provided, and equivalent disturbance compensation is introduced into the approximation law; constructing an ideal switching dynamic state based on an ellipse approach law, designing a controller according to the ideal switching dynamic state, and taking a signal obtained by current calculation as the input of a servo system; the specific controller parameter setting can be carried out according to the convergence performance index of the representation system, and a calculation formula of a monotone convergence layer boundary, an absolute convergence layer boundary and a quasi-sliding mode band boundary in the convergence process of the representation system is provided. The repetitive controller with the equivalent disturbance compensation ellipse approximation law can improve the tracking accuracy of a system and completely inhibit the existing periodic disturbance.

Description

Discrete repetitive control method of ellipse approximation law adopting disturbance compensation
Technical Field
The invention relates to a discrete repetitive control method of an ellipse approximation law by adopting disturbance compensation, which is used for accurately controlling the position of a motor servo system and is also suitable for other industrial occasions containing periodic operation processes.
Background
The controller for correcting the previous moment by the tracking error signal is repeatedly controlled to form the controller input of the current moment, so that the repeated control has the characteristics of completely inhibiting periodic interference and realizing accurate control. The current repetitive control method is widely applied to various high-precision servo motor driving systems.
The repetitive control technique is a control method based on inner membrane control. The so-called inner-envelope principle is that when a signal is considered as the output of an autonomous system and a model of the signal is "embedded" in a stable closed-loop system, the controlled output can track the reference signal completely. Therefore, designing the controller according to the inner-membrane principle requires constructing a periodic reference inner-membrane for one period T
Figure BDA0002622512510000011
It can be delayed by this period (e)-Ts) The positive feedback loop is realized, the specific form of the input signal is not considered, the internal module accumulates the input signal cycle by cycle as long as the initial section signal is given, and the signal with the same cycle as the previous cycle is repeatedly output.
The sliding mode controller of the deterministic system is designed by an approach law method, the design process is clear, and the parameter adjustment method of the controller is clear and easy to realize; however, for an uncertain system, the same approach law is adopted to design the controller, the switching dynamics caused by the design depends on an uncertain item, and the influence degree of the uncertain item on the switching dynamics determines the control performance of the system. Thus, the approach law needs to be modified to "embed" the interference suppression measures into the switching dynamics to obtain the ideal switching dynamics, so that the approach law method can be applied to uncertain systems. When designing a discrete controller, the indexes describing transient and steady-state behaviors of the tracking error can be given by ideal switching dynamics, and specifically, the following three indexes are provided: a monotonic convergence layer boundary, an absolute convergence layer boundary, and a quasi-sliding mode band boundary. In fact, the specific values of the three indexes depend on the controller parameters, the controller parameters are different, and the values of the three indexes are also different. Once the ideal switching dynamic form is given, specific expressions of three indexes can be given in advance for parameter setting of the controller.
Disclosure of Invention
In order to overcome the defects of the prior art, the invention provides a discrete repetitive control method for a motor servo system by adopting an ellipse approximation law of disturbance compensation. In order to make the closed loop system have the preset expected error tracking performance and effectively inhibit buffeting, a novel approach law, namely an ellipse approach law is provided, and a motor servo repetitive controller is designed according to the ideal switching dynamic of the approach law structure. The method has the advantages that the periodic interference components are completely inhibited, meanwhile, the non-periodic components existing in the disturbance are considered, the non-periodic interference is inhibited by introducing equivalent disturbance compensation in a closed-loop system, so that the control performance is improved, and the motor servo system can realize high-speed and high-precision tracking.
The technical scheme adopted by the invention for solving the technical problems is as follows:
a discrete repetitive control method of an ellipse approach law with perturbation compensation, the control method comprising the steps of:
1) differential equation model of servo system
e(k+1)＝Ae(k)+b(u(k)+w(k)) (1)
Wherein e (k), e (k +1) represents the tracking error of the servo system at the time k, k +1, u (k) represents the input control of the servo system at the time k, w (k) represents the interference signal of the servo system at the time k, A represents the servo system matrix, and b represents the control coefficient of the system;
2) given a periodic reference signal, satisfy
r(k)＝r(k-N) (2)
Wherein, N represents the period of the reference signal, r (k) represents the reference signal of the servo system at the k time, and r (k-N) represents the reference signal of the servo system at the k-th time of the last period;
3) defining tracking error
e(k)＝y(k)-r(k)
(3)
Wherein y (k) represents the system output of the servo system at time k;
4) selecting a linear switching function as s (k) cTe(k)，cTSelecting a gain parameter which determines the convergence and the convergence speed of the system on the sliding mode surface;
5) structural equivalent disturbance
d(k)＝cTb(w(k)-w(k-N)) (4)
Wherein d (k) represents an equivalent disturbance signal of the servo system at the k moment, and w (k-N) represents an interference signal of the servo system at the k-th moment of the last period;
6) constructing discrete time ellipse approximation law
s(k+1)＝(1-ρ)s(k)-fal(s(k),) (5)
Wherein s (k +1) represents a switching function at the time of k + 1; the elliptic function constructed in the approximation law is
Figure BDA0002622512510000021
Rho and rho are adjustable parameters, and rho is more than 0 and less than 1 and is more than 0 and more than 0;
7) constructing ideal switching dynamic according to the ellipse approximation law
s(k+1)＝(1-ρ)s(k)-fal(s(k),)+d(k)-d*(k) (6)
Wherein d is*(k) Represents the compensation amount at time d (k); note du，dlRespectively, upper and lower limits of d (k), d (k) satisfying du≥d(k)≥dlGet it
Figure BDA0002622512510000031
Get | d*(k) -d (k) ≦ Δ, where Δ is the bound of interference in the ideal switching dynamics equation (6);
8) and dynamically constructing a model of the discrete sliding mode repetitive controller according to the ideal switching:
u(k)＝u(k-N)+(cTb)-1[(1-ρ)s(k)-fal(s(k),)-s(k+1-N)-cTA(e(k)-e(k-N))-d*(k)] (7)
wherein u (k-N) represents a control variable of the servo system at the kth time of the previous period, and (c)Tb)-1To representMatrix coefficients, s (k +1-N) representing the switching function of the servo system at the k +1 th instant of the previous cycle, cTA represents a gain coefficient, and e (k-N) represents a tracking error variable of the servo system at the kth moment of the last period;
and (k) taking u (k) as a controller input signal of a servo object, measuring to obtain a servo system output signal y (k), and changing along with a reference signal r (k).
Further, after the design of the repetitive controller is completed, the parameter setting of the controller is carried out according to an index representing the convergence performance of the system, and in order to represent the convergence performance of the system, concepts of a monotone convergence layer boundary, an absolute convergence layer boundary and a quasi-sliding mode band boundary are introduced, and the specific definitions are as follows:
monotonic convergence layer boundary ΔMDR: outside the boundary of the monotonic convergence layer, s (k) decreases monotonically with the same sign, i.e.
Figure BDA0002622512510000032
Absolute convergence layer boundary ΔAL: outside the absolute convergence layer, | s (k) | monotonically decreases, i.e.
Figure BDA0002622512510000033
Quasi-slip form band boundary ΔQSM: the system switches over the plane s (k) ═ cTAn area of e (k) 0, which is stabilized once inside the area, i.e. a field in which the movement is stable
Figure BDA0002622512510000034
Phi monotonic convergence layer boundary DeltaMDR
Figure BDA0002622512510000041
Wherein
Figure BDA0002622512510000042
Figure BDA0002622512510000043
Absolute convergence layer boundary deltaAL
Figure BDA0002622512510000044
Wherein
Figure BDA0002622512510000045
Figure BDA0002622512510000046
③ boundary Delta of quasi-slip form beltQSM
Figure BDA0002622512510000047
(i)≤Z1
Figure BDA0002622512510000051
(ii)Z1＜＜Z2
Figure BDA0002622512510000052
(iii)Z1＜Z2＜
Figure BDA0002622512510000053
Wherein
Figure BDA0002622512510000054
Figure BDA0002622512510000055
Still further, the adjustable parameters of the controller include ρ, and; the parameter setting is based on the index representing the convergence process.
Further, in the step 5), let d be assumedu≥d(k)≥dlThen, then
Figure BDA0002622512510000056
The invention has the technical idea that the design of the discrete repetitive controller of the motor servo system is carried out based on the discrete time ellipse approximation law, is a time domain design method and is different from the currently and generally adopted frequency domain method. The given reference signal is considered when the controller is designed, and the designed controller is more visual, simple and convenient and is easy to depict the tracking performance of the system. The time domain design of the controller is easy to combine with the existing interference suppression means, the designed repetitive controller can realize the complete suppression of periodic interference signals and reduce the errors generated by non-periodic interference signals, and the quick and high-precision tracking of the given reference signals is realized.
The invention has the main effects that: the method has the advantages of quick convergence, accelerated interference suppression and higher control precision.
Drawings
Fig. 1 is a block diagram of an ac permanent magnet synchronous motor servo system.
FIG. 2 is a block diagram of a repetitive control controller for the ellipse approach law.
Fig. 3 is Δ when c is-0.5, ═ 0.1, ρ is 0.6, ═ 0.8, and Δ is 0.3MDR，ΔALAnd deltaQSMSchematic representation.
Fig. 4 shows the case where c is-0.5, 0.1, p is 0.5, and 0Δ when Δ is 0.3, 9MDR，ΔALAnd deltaQSMSchematic representation.
Fig. 5 to 12 are experimental results of the permanent magnet synchronous motor control system when the feedback controller parameter c is-0.5, ρ is 0.5, and 0.9, where:
FIG. 5 is a reference position signal and an actual position signal under the influence of a feedback controller based on an ellipse approximation law.
FIG. 6 is a controller voltage signal under the action of a feedback controller based on the ellipse approach law.
Fig. 7 is a graph of the switching function under the action of a feedback controller based on the ellipse approximation law.
Fig. 8 is a switching function distribution histogram under the action of a feedback controller based on an ellipse approximation law.
FIG. 9 is a reference position signal and an actual position signal under the influence of a feedback controller based on an elliptical approximation law and equivalent disturbance compensation.
FIG. 10 is a graph of the controller voltage signal under the influence of a feedback controller based on the elliptic approximation law and equivalent disturbance compensation.
FIG. 11 is a graph of the switching function under the action of a feedback controller based on the ellipse approximation law and equivalent disturbance compensation.
Fig. 12 is a switching function distribution histogram under the action of a feedback controller based on an ellipse approximation law and equivalent disturbance compensation.
Fig. 13 to 20 are experimental results of the permanent magnet synchronous motor control system when the repetitive controller parameter c is-0.5, ρ is 0.5, and 0.9, where:
FIG. 13 is a graph of a reference position signal and an actual position signal under the influence of a repetitive controller based on an ellipse approximation law.
FIG. 14 is a graph of the controller voltage signal under the influence of a repetitive controller based on the ellipse approximation law.
Fig. 15 is a graph of the switching function under the action of a repetitive controller based on the ellipse approximation law.
Fig. 16 is a switching function distribution histogram under the action of a repetitive controller based on an ellipse approximation law.
FIG. 17 is a reference position signal and an actual position signal under the influence of a repetitive controller based on an elliptical approximation law and equivalent disturbance compensation.
FIG. 18 is a graph of the controller voltage signal under the influence of a repetitive controller based on an elliptical approximation law and equivalent disturbance compensation.
FIG. 19 is a graph of the switching function under the influence of a repetitive controller based on the ellipse approximation law and equivalent disturbance compensation.
FIG. 20 is a histogram of the distribution of the switching function under the influence of a repetitive controller based on the ellipse approximation law and equivalent disturbance compensation.
Fig. 21 to 28 are experimental results of the permanent magnet synchronous motor control system when the feedback controller parameter c is-0.5, ρ is 0.7, 0.3, and 0.8, where:
fig. 21 shows a reference position signal and an actual position signal under the action of a feedback controller based on an ellipse approximation law.
FIG. 22 is a graph of the controller voltage signal under the influence of a feedback controller based on the ellipse approach law.
Fig. 23 is a graph of a switching function under the action of a feedback controller based on an ellipse approximation law.
Fig. 24 is a switching function distribution histogram under the action of a feedback controller based on an ellipse approximation law.
FIG. 25 is a graph of a reference position signal and an actual position signal under the influence of a feedback controller based on an elliptical approximation law and equivalent disturbance compensation.
FIG. 26 is a graph of the controller voltage signal under the influence of a feedback controller based on the elliptic approximation law and equivalent disturbance compensation.
FIG. 27 is a graph of the switching function under the influence of a feedback controller based on the ellipse approximation law and equivalent disturbance compensation.
Fig. 28 is a switching function distribution histogram under the action of a feedback controller based on the elliptic attraction law and equivalent disturbance compensation.
Fig. 29 to 36 are experimental results of the permanent magnet synchronous motor control system when the repetitive controller parameter c is-0.5, ρ is 0.7, 0.3, and 0.8, where:
fig. 29 shows a reference position signal and an actual position signal under the action of a repetitive controller based on the ellipse approximation law.
FIG. 30 is a graph of the controller voltage signal under the influence of a repetitive controller based on the ellipse approximation law.
Fig. 31 is a graph of the switching function under the action of a repetitive controller based on the ellipse approximation law.
Fig. 32 is a switching function distribution histogram under the action of a repetitive controller based on an ellipse approximation law.
FIG. 33 is a graph of a reference position signal and an actual position signal under the influence of a repetitive controller based on an elliptical approximation law and equivalent disturbance compensation.
FIG. 34 is a graph of the controller voltage signal under the influence of a repetitive controller based on the elliptic approximation law and equivalent disturbance compensation.
FIG. 35 is a graph of the switching function under the influence of a repetitive controller based on the ellipse approximation law and equivalent disturbance compensation.
FIG. 36 is a histogram of the distribution of the switching function under the influence of a repetitive controller based on the ellipse approximation law and equivalent disturbance compensation.
Detailed Description
The embodiments of the present invention will be further described with reference to the accompanying drawings.
Referring to fig. 1 and 2, a discrete repetitive control method of an ellipse approximation law using equivalent disturbance compensation, wherein fig. 1 is a block diagram of a motor servo system; FIG. 2 is a schematic diagram of a repetitive control controller for the ellipse approach law; the control method comprises the following steps:
1) differential equation model of servo system
e(k+1)＝Ae(k)+b(u(k)+w(k)) (1)
Wherein e (k), e (k +1) represents the tracking error of the servo system at the time k, k +1, u (k) represents the input control of the servo system at the time k, w (k) represents the interference signal of the servo system at the time k, A represents the servo system matrix, and b represents the control coefficient of the system;
2) given a periodic reference signal, satisfy
r(k)＝r(k-N) (2)
Wherein, N represents the period of the reference signal, r (k) represents the reference signal of the servo system at the k time, and r (k-N) represents the reference signal of the servo system at the k-th time of the last period;
3) defining tracking error
e(k)＝y(k)-r(k) (3)
Wherein y (k) represents the system output of the servo system at time k;
4) selecting a linear switching function as s (k) cTe(k)，cTSelecting a gain parameter which determines the convergence and the convergence speed of the system on the sliding mode surface;
5) structural equivalent disturbance
d(k)＝cTb(w(k)-w(k-N)) (4)
Wherein d (k) represents an equivalent disturbance signal of the servo system at the k moment, and w (k-N) represents an interference signal of the servo system at the k-th moment of the last period;
6) constructing discrete time ellipse approximation law
s(k+1)＝(1-ρ)s(k)-fal(s(k),) (5)
Wherein s (k +1) represents a switching function at the time of k + 1; the elliptic function constructed in the approximation law is
Figure BDA0002622512510000081
Rho and rho are adjustable parameters, and rho is more than 0 and less than 1 and is more than 0 and more than 0;
7) constructing ideal switching dynamic according to the ellipse approximation law
s(k+1)＝(1-ρ)s(k)-fal(s(k),)+d(k)-d*(k) (6)
Wherein d is*(k) Represents the compensation amount at time d (k); note du，dlRespectively, upper and lower limits of d (k), d (k) satisfying du≥d(k)≥dlGet it
Figure BDA0002622512510000091
Get | d*(k) -d (k) ≦ Δ, where Δ is the bound of interference in the ideal switching dynamics equation (6);
8) and dynamically constructing a model of the discrete sliding mode repetitive controller according to the ideal switching:
u(k)＝u(k-N)+(cTb)-1[(1-ρ)s(k)-fal(s(k),)-s(k+1-N)-cTA(e(k)-e(k-N))-d*(k)] (7)
wherein u (k-N) represents a control variable of the servo system at the kth time of the previous period, and (c)Tb)-1Representing the matrix coefficients, s (k +1-N) representing the switching function of the servo system at the k +1 th instant of the previous cycle, cTA represents a gain coefficient, and e (k-N) represents a tracking error variable of the servo system at the kth moment of the last period;
and (k) taking u (k) as a controller input signal of a servo object, measuring to obtain a servo system output signal y (k), and changing along with a reference signal r (k).
Further, after the design of the repetitive controller is completed, the parameter setting of the controller is carried out according to an index representing the convergence performance of the system, and in order to represent the convergence performance of the system, concepts of a monotone convergence layer boundary, an absolute convergence layer boundary and a quasi-sliding mode band boundary are introduced, and the specific definitions are as follows:
monotonic convergence layer boundary ΔMDR: outside the boundary of the monotonic convergence layer, s (k) decreases monotonically with the same sign, i.e.
Figure BDA0002622512510000092
Absolute convergence layer boundary ΔAL: outside the absolute convergence layer, | s (k) | monotonically decreases, i.e.
Figure BDA0002622512510000093
Quasi-slip form band boundary ΔQSM: the system switches over the plane s (k) ═ cTAn area of e (k) 0, which is stabilized once inside the area, i.e. a field in which the movement is stable
Figure BDA0002622512510000094
Phi monotonic convergence layer boundary DeltaMDR
Figure BDA0002622512510000095
Wherein
Figure BDA0002622512510000101
Figure BDA0002622512510000102
Absolute convergence layer boundary deltaAL
Figure BDA0002622512510000103
Wherein
Figure BDA0002622512510000104
Figure BDA0002622512510000105
③ boundary Delta of quasi-slip form beltQSM
Figure BDA0002622512510000106
(i)≤Z1
Figure BDA0002622512510000107
(ii)Z1＜＜Z2
Figure BDA0002622512510000111
(iii)Z1＜Z2＜
Figure BDA0002622512510000112
Wherein
Figure BDA0002622512510000113
Figure BDA0002622512510000114
Still further, the adjustable parameters of the controller include ρ, and; the parameter setting is based on the index representing the convergence process.
Further, in the step 5), let d be assumedu≥d(k)≥dlThen, then
Figure BDA0002622512510000115
Example (b): a discrete repetitive control method for a motor servo system by adopting an equivalent disturbance compensation ellipse approach law comprises the following steps:
first step, second order difference equation model of motor servo object
y(k+1)+a1y(k)+a2y(k-1)＝b1u(k)+b2u(k-1)+w(k)
Wherein y (k +1), y (k) and y (k-1) respectively represent output signals of the servo system at the moments k +1, k and k-1, u (k) and u (k-1) respectively represent input signals of the servo system at the moments k and k-1, w (k) represents interference signals at the moment k of the system, a1、a2、b1、b2Representing a system parameter of the servo system, and obtaining a value of the system parameter through parameter estimation;
given periodic reference signals, satisfy
r(k)＝r(k-N) (2)
Wherein, N represents the period of the reference signal, r (k) represents the reference signal of the servo system at the k time, and r (k-N) represents the reference signal of the servo system at the k-th time of the last period;
third, construct the discrete-time ellipse approximation law
s(k+1)＝(1-ρ)s(k)-fal(s(k),) (5)
Wherein the ellipse function in the approach law is
Figure BDA0002622512510000121
Rho and rho are adjustable parameters, and rho is more than 0 and less than 1 and is more than 0 and more than 0;
introducing equivalent disturbance to construct an ellipse approximation law with interference suppression effect
s(k+1)＝(1-ρ)s(k)-fal(s(k),)+d(k)-d*(k) (6)
Wherein d (k) represents an equivalent disturbance signal of the permanent magnet synchronous motor at the time k, d*(k) Represents the compensation amount at time d (k);
fifthly, a reference signal r (k) is given, and a system equation is written by a second-order difference equation model of a motor servo object
e(k+1)＝Ae(k)+b(v(k)+w(k)) (1)
Wherein e (k) ═ e1(k) e2(k)]T,e1(k)＝y(k-1)-e(k-1)，e2(k)＝y(k)-e(k)，e1(k) Tracking error at time k-1, e2(k) For tracking error at time k, input
Figure BDA0002622512510000122
For control inputs defined in the system tracking error equation, and the system matrix and control coefficients are respectively
Figure BDA0002622512510000123
The sixth step is to select the linear switching function as s (k) cTe (k), wherein c is selected as c [ -0.51 [ - ]]T；
Aiming at a second-order difference equation model of a motor servo object, in order to realize ideal switching dynamics, a discrete sliding mode repetitive controller is designed as follows:
Figure BDA0002622512510000131
for the above repetitive controller design, the following description is made:
1) e (k), y (k-1) and y (k-1-N) are all measured, and u (k-1) and u (k-1-N) represent stored values of the control signals, which can be read from the memory.
2) When the reference signal satisfies r (k) r (k-1), the discrete repetitive controller is also suitable for the constant value regulation problem, and the equivalent disturbance is d (k) w (k-1); wherein r (k-1) represents a reference signal of the servo system at the moment of k-1, and w (k-1) represents an interference signal of the servo system at the moment of k-1;
Figure BDA0002622512510000132
3) the above-mentioned repetitive controller can also give the design result of a higher-order system in the same way for a second-order system.
Eighth step, according to the monotone convergence layer boundary Delta of the convergence speed of the characterization systemMDRAbsolute convergence layer boundary ΔALAnd a quasi-slip form band boundary DeltaQSMAnd setting the parameters of the controller to achieve the optimal control effect. Wherein the controller parameters mainly include: an ellipse parameter, an adjustable parameter rho and an equivalent perturbation boundary delta;
according to the above-mentioned DeltaMDR、ΔALAnd deltaQSMThe determined boundary values are as follows:
monotonic convergence layer boundary ΔMDR
Figure BDA0002622512510000133
Absolute convergence layer boundary ΔAL
Figure BDA0002622512510000134
Quasi-slip form band boundary ΔQSM
Figure BDA0002622512510000141
Phi monotonic convergence layer boundary DeltaMDR
Figure BDA0002622512510000142
Wherein
Figure BDA0002622512510000143
Figure BDA0002622512510000144
Absolute convergence layer boundary deltaAL
Figure BDA0002622512510000145
Wherein
Figure BDA0002622512510000146
Figure BDA0002622512510000147
③ boundary Delta of quasi-slip form beltQSM
Figure BDA0002622512510000148
(i)≤Z1
Figure BDA0002622512510000151
(ii)Z1＜＜Z2
Figure BDA0002622512510000152
(iii)Z1＜Z2＜
Figure BDA0002622512510000153
Wherein
Figure BDA0002622512510000154
Figure BDA0002622512510000155
And calculating boundary values according to the formulas (11) to (16) to determine the tracking performance of the closed-loop system.
In this embodiment, for example, a servo system of a permanent magnet synchronous motor performs a repetitive tracking task on a fixed interval, and a position reference signal of the servo system has a periodically symmetric characteristic. TMS320F2812DSP is used as a controller, an AC servo motor APM-SB01AGN is used as a control object, and a permanent magnet synchronous motor servo system is formed by the AC servo driver of the ELMO and an upper mechanism to control the position of the motor. The servo system adopts three-loop control, the current loop and speed loop controller are provided by an ELMO driver, and the position loop is provided by a DSP development board.
The position loop controller is designed, and a mathematical model of a servo object except the position loop is required to be established, wherein the mathematical model comprises a current loop, a speed loop, a power driver, an alternating current permanent magnet synchronous servo motor body and a detection device. Obtaining a mathematical model of the servo object by parameter estimation
y(k+1)-1.8949y(k)+0.8949y(k-1)＝1.7908u(k)-0.5704u(k-1)+w(k) (17)
Wherein y (k), u (k) respectively represent the position output and the velocity given signal (control input) of the position servo system, and w (k) represents the interference signal.
Since the present embodiment uses a sinusoidal signal as the reference signal of the system, the repetitive controller may take the form of a controller given by equation (6), and a specific expression thereof may be written as
Figure BDA0002622512510000161
The effectiveness of the repetitive controller given by the present invention will be illustrated in this example by numerical simulation and experimental results.
Given a position reference signal of r (k) 10sin (2 pi fkT)s) In degrees (deg), frequency f 1Hz, and sampling period Ts0.001s, and the sampling period N is 1000. In simulation, the selected interference signal w (k) is composed of periodic disturbance and non-periodic random disturbance in the specific form of
w(k)＝2sin(2πfkTs)+0.15sgn(sin(2kπ/150)) (19)
Under the action of the repetitive controller (18), different controller parameters rho and the three boundary layers of the servo system are selected. For purposes of illustrating the invention patent with respect to monotonically converging layer boundaries ΔMDRAbsolute convergence layer boundary ΔALAnd quasi-slip form band boundary deltaQSMThe theoretical correctness of (1) and numerical simulation are carried out.
1) When the controller parameter c is-0.5, ═ 0.1, ═ 0.6, ═ 0.8, and Δ is 0.3, the calculation formula for the three boundary values yields
ΔMDR＝0.9926，ΔAL＝0.6642，ΔQSM＝0.3145
2) When the controller parameter c is-0.5, ═ 0.1, (-) 0.5, (-) 0.9, and Δ is 0.3, the calculation formula for the three boundary values yields
ΔMDR＝ΔAL＝0.7987，ΔQSM＝0.3109
The simulation results are shown in fig. 3-4. At a given system model, reference signalAnd in the case of interference signals, the numerical results verify the monotonous convergence layer boundary Delta of the system for representing the convergence speed under the action of the repetitive controller provided by the patentMDRAbsolute convergence layer boundary ΔALAnd quasi-slip form band boundary deltaQSMThe accuracy of (2).
The block diagram of the permanent magnet synchronous motor control system used in the experiment is shown in figure 1. And verifying the tracking performance of discrete repetitive control based on the ellipse approximation law by setting different controller parameters. Given a position signal of rk＝A(sin(2πfkTs) +1), where the amplitude a is 135deg, the sampling period Ts5ms and 1 Hz.
The feedback controller is adopted to take the following form
Figure BDA0002622512510000171
The feedback controller based on disturbance compensation is adopted in the following form
Figure BDA0002622512510000172
The repetitive controller is adopted in the following form
Figure BDA0002622512510000173
The repetitive controller based on disturbance compensation is adopted in the following form
Figure BDA0002622512510000174
A. The controller parameters are c-0.5, ρ -0.5, 0.9, the reference signal period T-4 s, and the sampling period Ts-0.005 s.
Using the feedback controller (20), the position signal, controller voltage, switching function curve, and switching function histogram are shown in FIGS. 5-8, where Δ isQSM＝0.12deg。
B. The controller parameters are c-0.5, ρ -0.5, 0.9, the reference signal period T-4 s, and the sampling period Ts-0.005 s.
Using a feedback controller (21) based on disturbance compensation, the position signal, controller voltage, switching function curve and switching function histogram are shown in FIGS. 9-12, where Δ isQSM＝0.1deg。
C. The controller parameters are c-0.5, ρ -0.5, 0.9, the reference signal period T-4 s, and the sampling period Ts-0.005 s.
With repetitive controllers (22), the position signals, controller voltages, switching function curves and switching function histograms are shown in FIGS. 13-16, where Δ isQSM＝0.07deg。
D. The controller parameters are c-0.5, ρ -0.5, 0.9, the reference signal period T-4 s, and the sampling period Ts-0.005 s.
With repetitive controllers (23) based on disturbance compensation, the position signal, controller voltage, switching function curve and switching function histogram are shown in FIGS. 17-20, where Δ isQSM＝0.05deg。
E. The controller parameters are c-0.5, ρ -0.7, 0.3, 0.8, the reference signal period T-4 s, and the sampling period Ts-0.005 s.
Using the feedback controller (20), the position signal, controller voltage, switching function curve, and switching function histogram are shown in FIGS. 21-24, where Δ isQSM＝0.11deg。
F. The controller parameters are taken as c-0.5, ρ -0.7, 0.3, 0.8, the reference signal period T-4 s, and the sampling period Ts-0.005 s.
Using a feedback controller (21) based on disturbance compensation, the position signal, controller voltage, switching function curve and switching function histogram are shown in FIGS. 25-28, where Δ isQSM＝0.09deg。
G. The controller parameters are c-0.5, ρ -0.7, 0.3, 0.8, the reference signal period T-4 s, and the sampling period Ts-0.005 s.
Using a repetitive controller (22), position signals, controller voltages, switching function curves andthe histogram of the switching function is shown in FIGS. 29-32, where Δ isQSM＝0.07deg。
H. The controller parameters are c-0.5, ρ -0.7, 0.3, 0.8, the reference signal period T-4 s, and the sampling period Ts-0.005 s.
With repetitive controllers (23) based on disturbance compensation, the position signal, controller voltage, switching function curve and switching function histogram are shown in FIGS. 33-36, where Δ isQSM＝0.05deg。
I. The controller parameters are c-0.5, ρ -0.8, 0.1, 0.7, the reference signal period T-4 s, and the sampling period Ts-0.005 s.
The experimental results show that the ellipse approximation law provided by the invention can effectively inhibit buffeting during system track tracking; equivalent disturbance is introduced, and equivalent disturbance compensation is used as compensation of modeling characteristics and external unknown disturbance, so that unknown disturbance is effectively suppressed; and the periodic disturbance is completely inhibited by combining with repeated control, so that the control performance of the system is further improved.

Claims (4)

1. A discrete repetitive control method of an ellipse approach law adopting disturbance compensation is characterized in that: the control method comprises the following steps:
1) differential equation model of servo system
e(k+1)＝Ae(k)+b(u(k)+w(k)) (1)
Wherein e (k), e (k +1) represents the tracking error of the servo system at the time k, k +1, u (k) represents the input control of the servo system at the time k, w (k) represents the interference signal of the servo system at the time k, A represents the servo system matrix, and b represents the control coefficient of the system;
2) given a periodic reference signal, satisfy
r(k)＝r(k-N) (2)
Wherein, N represents the period of the reference signal, r (k) represents the reference signal of the servo system at the k time, and r (k-N) represents the reference signal of the servo system at the k-th time of the last period;
3) defining tracking error
e(k)＝y(k)-r(k) (3)
Wherein y (k) represents the system output of the servo system at time k;
4) selecting a linear switching function as s (k) cTe(k)，cTSelecting a gain parameter which determines the convergence and the convergence speed of the system on the sliding mode surface;
5) structural equivalent disturbance
d(k)＝cTb(w(k)-w(k-N)) (4)
Wherein d (k) represents an equivalent disturbance signal of the servo system at the k moment, and w (k-N) represents an interference signal of the servo system at the k-th moment of the last period;
6) constructing discrete time ellipse approximation law
s(k+1)＝(1-ρ)s(k)-fal(s(k),) (5)
Wherein s (k +1) represents a switching function at the time of k + 1; the elliptic function constructed in the approximation law is
Figure FDA0002622512500000011
Rho and rho are adjustable parameters, and rho is more than 0 and less than 1 and is more than 0 and more than 0;
7) constructing ideal switching dynamic according to the ellipse approximation law
s(k+1)＝(1-ρ)s(k)-fal(s(k),)+d(k)-d*(k) (6)
Wherein d is*(k) Represents the compensation amount at time d (k); note du，dlRespectively, upper and lower limits of d (k), d (k) satisfying du≥d(k)≥dlGet it
Figure FDA0002622512500000021
Get | d*(k) -d (k) ≦ Δ, where Δ is the bound of interference in the ideal switching dynamics equation (6);
8) according to the ideal switching dynamic construction discrete sliding mode repetitive controller
u(k)＝u(k-N)+(cTb)-1[(1-ρ)s(k)-fal(s(k),)-s(k+1-N)-cTA(e(k)-e(k-N))-d*(k)] (7)
Wherein u (k-N) represents a servoThe control variable of the system at the kth time of the previous cycle, (c)Tb)-1Representing the matrix coefficients, s (k +1-N) representing the switching function of the servo system at the k +1 th instant of the previous cycle, cTA represents a gain coefficient, and e (k-N) represents a tracking error variable of the servo system at the kth moment of the last period;
and (k) taking u (k) as a controller input signal of a servo object, measuring to obtain a servo system output signal y (k), and changing along with a reference signal r (k).
2. The discrete repetitive control method using the disturbance-compensated ellipse approximation law as claimed in claim 1, wherein: after the design of the repetitive controller is finished, the parameter setting of the controller is carried out according to the index representing the convergence performance of the system; in order to represent the convergence performance of a system, concepts of a monotone convergence layer boundary, an absolute convergence layer boundary and a quasi-sliding mode band boundary are introduced, and the concepts are specifically defined as follows:
monotonic convergence layer boundary ΔMDR: outside the boundary of the monotonic convergence layer, s (k) decreases monotonically with the same sign, i.e.
Figure FDA0002622512500000022
Absolute convergence layer boundary ΔAL: outside the absolute convergence layer, | s (k) | monotonically decreases, i.e.
Figure FDA0002622512500000023
Quasi-slip form band boundary ΔQSM: the system switches over the plane s (k) ═ cTAn area of e (k) 0, which is stabilized once inside the area, i.e. a field in which the movement is stable
Figure FDA0002622512500000024
Phi monotonic convergence layer boundary DeltaMDR
Figure FDA0002622512500000031
Wherein
Figure FDA0002622512500000032
Figure FDA0002622512500000033
Absolute convergence layer boundary deltaAL
Figure FDA0002622512500000034
Wherein
Figure FDA0002622512500000035
Figure FDA0002622512500000036
③ boundary Delta of quasi-slip form beltQSM
Figure FDA0002622512500000037
(i)≤Z1
Figure FDA0002622512500000041
(ii)Z1＜＜Z2
Figure FDA0002622512500000042
(iii)Z1＜Z2＜
Figure FDA0002622512500000043
Wherein
Figure FDA0002622512500000044
Figure FDA0002622512500000045
3. A discrete repetitive control method using disturbance-compensated ellipse approximation law as claimed in claim 1 or 2, wherein: the adjustable parameters of the controller include ρ, and; the parameter setting is based on the index representing the convergence process.
4. A discrete repetitive control method using disturbance-compensated ellipse approximation law as claimed in claim 1 or 2, wherein: in said step 5), let d be assumedu≥d(k)≥dlThen, then
Figure FDA0002622512500000046
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